Dynamical billiards constitute a very natural class of Hamiltonian systems: in
1927 George Birkhoff conjectured that, among all billiards inside smooth
planar convex domains, only billiards in ellipses are integrable. In this talk
we will prove a version of this conjecture for convex domains that are
sufficiently close to an ellipse of small eccentricity.
We will also describe some remarkable relation with inverse spectral theory
and spectral rigidity of planar convex domains. Our techniques can in fact be
fruitfully adapted to prove spectral rigidity among generic (finitely) smooth
axially symmetric domains which are sufficiently close to a circle. This gives
a partial answer to a question by P. Sarnak.